| L(s) = 1 | − 2·3-s − 3·5-s + 3·7-s + 3·9-s + 4·11-s + 6·15-s + 17-s − 6·19-s − 6·21-s − 4·23-s + 25-s − 4·27-s + 29-s − 31-s − 8·33-s − 9·35-s + 11·37-s + 41-s − 5·43-s − 9·45-s − 3·49-s − 2·51-s + 11·53-s − 12·55-s + 12·57-s + 14·59-s + 16·61-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 1.34·5-s + 1.13·7-s + 9-s + 1.20·11-s + 1.54·15-s + 0.242·17-s − 1.37·19-s − 1.30·21-s − 0.834·23-s + 1/5·25-s − 0.769·27-s + 0.185·29-s − 0.179·31-s − 1.39·33-s − 1.52·35-s + 1.80·37-s + 0.156·41-s − 0.762·43-s − 1.34·45-s − 3/7·49-s − 0.280·51-s + 1.51·53-s − 1.61·55-s + 1.58·57-s + 1.82·59-s + 2.04·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 65804544 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 65804544 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.201506578\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.201506578\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | $C_1$ | \( ( 1 + T )^{2} \) |
| 13 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 + 3 T + 8 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - 3 T + 12 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 17 | $D_{4}$ | \( 1 - T + 30 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 6 T + 30 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 29 | $D_{4}$ | \( 1 - T + 20 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + T + 58 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 11 T + 100 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - T + 78 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 5 T + 88 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 11 T + 98 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 14 T + 150 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 16 T + 169 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 5 T + 136 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 73 | $D_{4}$ | \( 1 + 12 T + 165 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 15 T + 210 T^{2} + 15 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 10 T + 174 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 18 T + 242 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 13 T + 232 T^{2} + 13 p T^{3} + p^{2} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.896800320392772036708691283164, −7.66341401072475005606625450741, −7.29798683280853323330037259825, −6.99723379377983517710511435693, −6.56748112265990479895797823892, −6.31490818844289264549754182433, −5.90052699246625139778732580003, −5.61343376550136172372785565258, −5.22991447078766990651523019076, −4.69192884917989858401857642703, −4.38466829270105116936186900874, −4.23690956725633125627596916430, −3.80713301148130194502081289730, −3.69234183090953880527652852295, −2.82706628027420485498912758518, −2.44477806279464849356551792791, −1.66660705026382484610041000020, −1.59431596078082280046255954366, −0.826134559964088491033833671505, −0.36867712249384187583558968223,
0.36867712249384187583558968223, 0.826134559964088491033833671505, 1.59431596078082280046255954366, 1.66660705026382484610041000020, 2.44477806279464849356551792791, 2.82706628027420485498912758518, 3.69234183090953880527652852295, 3.80713301148130194502081289730, 4.23690956725633125627596916430, 4.38466829270105116936186900874, 4.69192884917989858401857642703, 5.22991447078766990651523019076, 5.61343376550136172372785565258, 5.90052699246625139778732580003, 6.31490818844289264549754182433, 6.56748112265990479895797823892, 6.99723379377983517710511435693, 7.29798683280853323330037259825, 7.66341401072475005606625450741, 7.896800320392772036708691283164
